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Banach fixed point theorem for fractional integral contraction.

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Title: Banach fixed point theorem for fractional integral contraction.
Authors: Ayoob, Irshad1 (AUTHOR) iayoub@psu.edu.sa
Source: Fixed Point Theory & Algorithms for Sciences & Engineering. 12/22/2025, Vol. 2025 Issue 1, p1-16. 16p.
Subjects: Fixed point theory, Contraction operators, Metric spaces, Fractional integrals
Abstract: One of the most intensively studied and generalized results in metric fixed point theory is Branciari's fixed point theorem, which asserts that in a complete metric space (M , ρ) , a mapping T : M → M satisfying ∫ 0 ρ (T x , T y) ω (t) d t ≤ c ∫ 0 ρ (x , y) ω (t) d t for some ω : [ 0 , ∞) → [ 0 , ∞) , c ∈ (0 , 1) , and all x , y ∈ M , admits a unique fixed point x ∗ ∈ M. This reduces to Banach fixed theorem when ω (t) = 1. We extend this to Riemann–Liouville fractional integral contractions, proving the existence of a fixed point for T under 1 Γ (α) ∫ 0 ρ (T x , T y) (ρ (T x , T y) − t) α − 1 φ (t) d t ≤ c ⋅ 1 Γ (α) ∫ 0 ρ (x , y) (ρ (x , y) − t) α − 1 φ (t) d t , which recovers Branciari's integral condition for α = 1. [ABSTRACT FROM AUTHOR]
Copyright of Fixed Point Theory & Algorithms for Sciences & Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: One of the most intensively studied and generalized results in metric fixed point theory is Branciari's fixed point theorem, which asserts that in a complete metric space (M , ρ) , a mapping T : M → M satisfying ∫ 0 ρ (T x , T y) ω (t) d t ≤ c ∫ 0 ρ (x , y) ω (t) d t for some ω : [ 0 , ∞) → [ 0 , ∞) , c ∈ (0 , 1) , and all x , y ∈ M , admits a unique fixed point x ∗ ∈ M. This reduces to Banach fixed theorem when ω (t) = 1. We extend this to Riemann–Liouville fractional integral contractions, proving the existence of a fixed point for T under 1 Γ (α) ∫ 0 ρ (T x , T y) (ρ (T x , T y) − t) α − 1 φ (t) d t ≤ c ⋅ 1 Γ (α) ∫ 0 ρ (x , y) (ρ (x , y) − t) α − 1 φ (t) d t , which recovers Branciari's integral condition for α = 1. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Fixed Point Theory & Algorithms for Sciences & Engineering is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1186/s13663-025-00819-z
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      – SubjectFull: Metric spaces
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